Applied Energy 119 (2014) 371–383
Contents lists available at ScienceDirect
Applied Energy
journal homepage: www.elsevier.com/locate/apenergy
Design of Pareto optimal CO2 cap-and-trade policies for deregulated
electricity networks
Felipe Feijoo ⇑,1, Tapas K. Das 2
University of South Florida, 4202 E. Fowler Avenue, ENB118, Tampa, FL 33620, United States
h i g h l i g h t s
A mathematical–statistical model for designing Pareto optimal CO2 cap-and-trade policies.
The model fills a gap in the current literature that primarily supports cap-and-trade policy evaluation but not policy design.
Pareto optimal policies accommodate conflicting goals of the market constituents.
Electricity demand-price sensitivity and social cost of carbon have significant influence on the cap-and-trade policies.
Higher demand-price sensitivity increases the influence of penalty and social cost of carbon on reducing carbon emissions.
a r t i c l e
i n f o
Article history:
Received 15 August 2013
Received in revised form 2 January 2014
Accepted 4 January 2014
Keywords:
Electricity networks
Cap-and-trade
Game theory
MPEC/EPEC
a b s t r a c t
Among the CO2 emission reduction programs, cap-and-trade (C&T) is one of the most used policies. Economic studies have shown that C&T policies for electricity networks, while reducing emissions, will likely
increase price and decrease consumption of electricity. This paper presents a two layer mathematical–
statistical model to develop Pareto optimal designs for CO2 cap-and-trade policies. The bottom layer
finds, for a given C&T policy, equilibrium bidding strategies of the competing generators while maximizing social welfare via a DC optimal power flow (DC-OPF) model. We refer to this layer as policy evaluation.
The top layer (called policy optimization) involves design of Pareto optimal C&T policies over a planning
horizon. The performance measures that are considered for the purpose of design are social welfare and
the corresponding system marginal price (MP), CO2 emissions, and electricity consumption level.
Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction
A major part of the total CO2 emissions come from the electricity production sector, e.g., 40% in the U.S. ([1]). In 2009, 70% of the
electricity was produced from fossil fuel such as gas, coal, and
petroleum ([2]). In 2005, the European Union Emissions Trading
System (EU ETS) launched a cap-and-trade system that seeks to reduce the greenhouse gas (GHG) emissions by 21% by 2020 from the
2005 level. Currently, the EU ETS is the largest emission market in
the world [3], and according to the European Commission [4], at
least 20% of its budget for 2014–2020 will be spent on climate-related projects and policies. In the United States, as well as in the
EU, different regulations have been discussed to cut CO2 emissions
such as carbon tax, renewable portfolio standards (RPS), and cap-
⇑ Corresponding author. Tel.: +1 81 331 29776.
E-mail addresses: [email protected] (F. Feijoo), [email protected] (T.K. Das).
Industrial and Management Systems Engineering, University of South Florida,
United States.
2
Professor and Chair of Industrial and Management Systems Engineering, University of South Florida, United States.
1
0306-2619/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.apenergy.2014.01.019
and-trade programs (C&T). In the northeastern U.S., the Regional
Greenhouse Gas Initiative (RGGI) has already implemented a C&T
program through a nine state collaborative effort, which seeks to
cut the CO2 emissions by 10% by 2018. Recently the California
Air Resources Board adopted a C&T program, held its first auction
on November 2012, and started operation on January 2013. It is a
part of California’s historic climate change law (AB 32) that will reduce the carbon pollution to 1990 levels by 2020. Other countries
who have already implemented C&T programs include New Zealand, Japan, The Netherlands, and Australia.
C&T policies implemented in the past for greenhouse gases had
resulted in increase in cost for households, since the generators
passed the emissions reduction costs to the consumers [5]. Linn
[6] studied the economic impact of C&T policies for nitrogen oxides
(NOx ) on deregulated firms. The study showed that the effect of
reduction of NOx emissions may not increase electricity prices to
a level which could fully compensate firms for their compliance
costs, particularly for coal generators. This lead them to reduce
their expected profit by as much as $25 billions. Chen et al. [7]
developed a mathematical model to examine the ability of larger
producers in an electricity market under NOx C&T policy to manip-
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
ulate both the electricity and emission allowances markets. With
regards to CO2 cap-and-trade policies, Ruth et al. [8] examined
the economic and energy impacts that the state of Maryland will
have by joining the RGGI initiative. They identified several issues
that are important to the acceptance and effectiveness of CO2
C&T programs, such as rules for allowances distribution and subsidies for energy efficiency programs. Bird et al. [9] claimed that,
while renewable energy will likely benefit from carbon cap-andtrade programs, C&T can also impact the ability of renewable energy generation to affect overall CO2 emissions levels. They summarized the key issues for markets that are emerging under CO2
C&T policies and also the policy design options to allow renewable
energy generation to impact the emissions level. For further reading about economic impacts of CO2 C&T, we refer the readers to
Goettle and Fawcett [10] and Parmesano and Kury [11]. Linares
et al. [12] have studied the impact of CO2 C&T regulation and green
certificates on power generation expansion model. They have
shown that when a C&T policy is considered, locational marginal
prices (LMPs) increase, emissions decrease, and installed green
generation capacity increases. Limpaitoon et al. [2] studied the impact of C&T regulation and the interaction of demand elasticity,
transmission network, market structure, and strategic behavior
for an oligopoly electricity market (in the state of California). They
concluded that GHG regulations will affect the system operations
and market outcomes by increasing electricity prices (for both oligopoly and perfect competition scenarios), and reducing both GHG
emissions and energy consumption. Their results also suggest that
the interaction between CO2 C&T regulations, market structure,
and congestion might lead to potential abuses of market power
and create more congestion, which will limit nuclear access to
the market. This generates higher demand for emissions permit,
which in turn increases permit prices. Fullerton and Metcalf [13]
showed how certain environmental policies reduce profit under
monopoly, raise prices, and reduce welfare profit. Rocha and Das
[14] presented a game-theoretic model for developing joint bidding strategies in C&T allowances and electricity markets for competing generators. Rocha et al. [15] examined the impact of C&T
policies on generation capacity investment.
Some of the current research concerning C&T policies is focused
on the issue of development and analysis of allowance allocation
mechanisms [16–18], and suggestions for policy effectiveness
[8,9,19]. However, the open literature does not offer a methodology for design of CO2 C&T policies that takes into account a range
of conflicting measures of performance that appeal to different
market stakeholders. For example, a consumer’s concern is price
increase [6,8,13], a generator’s concern is profit/revenue reduction
[2,13], and a policy maker’s concerns include adequate emissions
reduction and sustaining electricity consumption necessary to support economic growth [9,12,13]. There does not appear to be a consensus in the literature about economic impact that can be
expected under C&T programs, neither is there an agreement about
the choice of the C&T parameters and their values [20]. This paper
attempts to fill the above gaps by presenting a 2-layer mathematical–statistical model to design Pareto optimal CO2 C&T policies for
deregulated electricity networks. This paper presents an elaborate
sensitivity analysis of selected policy parameters (initial allowance
cap, cap reduction rate, violation penalty) and network parameters
(congestion, social cost of carbon, and demand-price sensitivity of
the consumers).
In the bottom layer of the 2-layer model, the strategic bidding
behavior of the competing generators is formulated as a bi-level
mathematical model. The upper level model focuses on maximizing overall generator profit by bidding to the independent system
operator (ISO) in the allowance and electricity markets. The lower
level model focuses on social welfare maximization (or, social cost
minimization) while meeting the network and policy constraints
via a DC-OPF model. Each bi-level optimization problem is reformulated as an mathematical problem with equilibrium constraints
(MPEC). Equilibrium bidding strategies among the competing generators are obtained by solving the set of MPECs as an equilibrium
problem with equilibrium constraints (EPEC). Thus, the bottom
layer of the 2-layer model essentially evaluates the impact of a given C&T policy on a network by obtaining the performance measures including electricity price, emissions level, and
consumption level. The top layer model, using as input the results
of the bottom layer model, develops regression equations for different network performance measures using the tools of analysis
of variance (ANOVA). The regression equations relate the performance measures to the parameters of the C&T policy and the network. These equations are used in forming a multi-objective
optimization model, solution of which yields the Pareto optimal
CO2 C&T policy designs.
The rest of the paper is organized as follow. In Section 2 we
introduce the elements of a C&T policy and discuss social cost of
carbon. Section 3 presents the complete 2-layer model-based
methodology that obtains the Pareto optimal C&T policies. Section 4 demonstrates the application of our methodology on a sample network. Section 5 provides the concluding remarks.
2. Cap-and-trade and the social cost of carbon
Cap-and-trade is a market based mechanism that can be used to
regulate the GHG emissions. The following are some of the primary
features of a cap-and-trade mechanism.
Point of regulation: Different approaches to regulate emissions
in the electricity markets have been proposed and implemented. They range from regulating far upstream at the point
of sale of fossil fuels to far downstream at the point of purchase
of manufactured products and energy by ultimate consumers
[21]. The upstream approach sets an emissions cap on producers of raw material that contains GHG (e.g., coal, gas, or petroleum), whereas the downstream approach regulates the direct
producers of GHG [22].
Allowance distribution: In an Emissions Trading System (ETS),
one of the major concerns is how to distribute the allowances,
as both the initial as well as the continuing distribution strategies have a significant influence on the final market equilibrium.
Under a downstream regulation, several allowance allocation/
distribution mechanisms have been studied in recent years.
Most commonly discussed mechanisms in the literature are free
allocation and auction based allocation [18,17]. Free allocation
of allowances is often based either on historical emissions
(known as grandfathering) or on energy input/product output
(known as benchmarking) [23]. An allocation mechanism based
on equal per capita cumulative emissions was presented in [24].
RGGI has implemented an auction based model with a combination of uniform and discriminatory pricing strategies. In the
EU ETS, during the first and second trading periods, most of
the allowances were given freely according to historical emissions. In the third trading period, free allowances allocation is
scheduled to be progressively replaced by auctioning through
2020 [23].
Cap stringency: This is a common feature in all C&T policies and
refers to the rate of reduction of allowance cap. The cap reduction rate varies among markets due to network (market) intrinsic characteristics, policy decisions, external economic
variables, and other C&T parameter values [25]. For instance,
EU ETS implemented a 1.74% linear cap reduction for 2012–
2020 and beyond ([26]), while RGGI implemented a fixed cap
for 2009–2014 and a subsequent 2.5% reduction until 2018.
F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
Banking: It allows generators to save allowances for future periods. Some markets also allow secondary trading of excess
allowances at the end of each period. Banking rules could vary
among different markets. EU ETS permitted allowance banking
during phase I of its operation, but did not permit allowances to
be carried over into phase II. From phase II and on, unlimited
banking and borrowing is allowed [25]. The RGGI allows banking with no restrictions from current compliance periods into
the future.
Penalty: Fossil fuel generators must procure allowances commensurate with their emissions level or pay a penalty for any
shortfall. EU ETS introduced a monetary penalty (40 euros during phase I and 100 euros for phase II), whereas California’s policy require four additional allowances for each shortfall. IETA
(International Emissions Trading Association) advises against
using a non-compliance penalty by additional withdrawal of
allowances. Instead, IETA recommends adopting a more traditional fixed monetary fine similar to those entered in the SO2
and NOx emissions markets in the U.S. [27].
Other features of a cap-and-trade mechanism include initial cap
size, safety value, revenue recycling and cost containment mechanisms [28,29].
In this paper, we consider a downstream regulation for C&T. The
electricity generators are assume to acquire allowances by competing in an allowance market that is settled via discriminatory auction. The generators submit price-quantity bids that are arranged
in a decreasing price order by the auctioneer. The price paid by
each generator selected to received allowances is its own bid price
[17]. As presented in our model in Section 3.2, the independent
system operator (ISO) obtains the allowance allocation together
with generation dispatch by incorporating the allowances auction
model within the OPF model.
2.1. Social cost of carbon
The fiscal impact of CO2 emissions on the environment and
society is often referred to as the social cost of carbon (SCC). The
most common means that are used to characterize SCC include
marginal social cost of emissions and shadow prices of policies
(e.g., cap-and-trade). Mandell [32] presents various definitions
and estimates of SCC. The most common approach used in the literature defines the marginal social cost of carbon as ‘‘the cost to
the society for each additional unit of carbon (in the form of CO2 )
into the atmosphere.’’ However, as the carbon remains in the atmosphere, it is difficult to estimate the cost of the carbon in the future.
This motivates to redefine the SCC as ‘‘the present value of the
monetized damage caused by each period of emitting one extra
ton of CO2 today as compared to the baseline.’’ Another challenge
for assessing SCC is in estimating the discount rate of the future social cost of CO2 .
The estimation of the SCC will always suffer from uncertainty,
speculation, and lack of information [30]. This can be noticed, for
example, in (1) estimating future emissions of greenhouse gases,
(2) monetizing the effects of past and future emissions on the climate system, (3) assessing the impact of changes in climate on the
physical and biological environment, and (4) translating environmental impacts into economic damages. Therefore, any effort to
quantify and monetize the harms associated with climate change
is bound to raise serious questions of science, economics, and ethics, and thus should be viewed as provisional [30]. Avato et al. [31]
consider the carbon emissions as one of the main barriers to the
development and deployment of clean energy technologies. They
argue that emissions (a negative externality) is not valued and
therefore not included into investment decisions by energy providers. Hence, by considering the monetary effects of emissions, i.e.,
373
the SCC, an increase in development and deployment of green
technologies can be achieved.
Mandell [32] summarized the result of 211 SCC estimates from
47 different studies using an integrated assessment model (IAM).
The SCC was estimated to have a mean value of €19:70=tCO2 , a
median of €5:45=tCO2 and an estimated range from €1:24=tCO2
to €451=tCO2 . Using the shadow price approach, the estimated value of SCC has a range of €32=tCO2 to €205=tCO2 among the countries in the European Union. Hope [33] estimated SCC considering
two scenarios: low emissions case, and a business as usual
(BAU) emissions scenario. This study concluded that the median
SCC for the BAU scenario is $100=tCO2 with a range of
$10=tCO2 —$270=tCO2 , and a median of $50=tCO2 for the low emission scenario with a range of $5=tCO2 $130=tCO2 . The U.S. government, through the interagency working group (IWG),
calculated the cost imposed on the global society by each additional ton of CO2 . They included health impact, economic dislocation, agricultural changes and other effects that climate change
can impose on humanity. They estimated the SCC to have a range
from $5=tCO2 to $65=tCO2 . The IWG suggests setting the SCC to
$21=tCO2 . The IWG also propose to utilize a discount rate of 2.5–
5% to address future SCC [34].
3. A model for developing Pareto optimal C&T policies
In this section, we present the complete methodology for
designing Pareto optimal cap-and-trade policies for a fixed planning horizon. The methodology comprises a 2-layer model and a
detailed solution approach.
3.1. A 2-layer model
The mathematical–statistical model for obtaining Pareto optimal C&T policy designs has two broad layers, which we call the
top and the bottom layers.
The bottom layer (also referred to in this paper as the policy
evaluation layer) involves obtaining equilibrium bidding strategies
of the competing generators as a function of the C&T and network
parameters, while maximizing social welfare via a DC-OPF. The
network performance measures (emissions, electricity price, and
consumption level) corresponding to the equilibrium bidding
strategy, for a given C&T policy, are fed to the top layer (policy optimization layer) that obtains the Pareto optimal C&T policy designs.
The top layer model first performs an analysis of variance (ANOVA)
using the C&T and network parameters as factors, and the values of
the network performance measures from the bottom layer as the
responses. The ANOVA results are used to formulate regression
equations for the network performance measures as functions of
the C&T and network parameters. The Pareto optimal policy designs are then obtained by optimizing the regression equations
via a multi-objective optimization model. The 2-layer model is given as follows.
Optimize f ðc; r; p; b; p; lÞ
8
>
< Maximize g i ðai ; xi Þ; 8i 2 I
Bottom layer
s:t: Maximize WðQ ; HÞ
>
:
s:t: Network and policy constraints:
Top layerf
ð1Þ
In the top layer, f ðc; r; p; b; p; lÞ represents the objective function
of the multi-objective optimization model, where c is the cap size
representing the maximum tons of CO2 allowed per year (tCO2 =yr),
r is the yearly cap reduction rate (% of c) which goes into effect
after a preset number of initial years, p is the penalty for exceeding
the emissions limit per allocated allowances ($=tCO2 ), b represents
the demand price sensitivity (slope of the demand curve), p is the
social cost of carbon ($=tCO2 ), and l denotes the vector of line
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
capacities (MW h) in the network (a determinant for network congestion). The bottom layer is formulated, for each generator
i 2 I ¼ f1; 2; . . . ; ng, as a bi-level optimization model, where
g i ðai ; xi Þ denotes the profit of generator i, and ai (intercept of supply function) and xi (allowance price) are the generator bids in the
electricity and allowances markets, respectively. WðQ ; HÞ denotes
the social welfare objective function for the DC optimal power flow
problem, where Q ¼ ðq1 ; . . . ; qi ; . . . ; qn Þ is the electricity dispatch
and H ¼ ðh1 ; . . . ; hi ; . . . ; hn Þ is the allowances allocation. For each
generator i, the bi-level model is solved as a mathematical problem
with equilibrium constraints (MPEC). The equilibrium bidding
strategies for all generators in both electricity and allowances markets are obtained by solving an equilibrium problem with equilibrium constraints (EPEC).
A schematic of the solution methodology for the 2-layer model
is shown in Fig. 1. The methodology begins by designing a factorial
experiment with all C&T and network related parameters as factors
at two or more levels, and enumerating all possible factor level
combinations [box (1)]. Factor level combinations, one at a time,
while yet to be evaluated [box (2)], are forwarded as input to the
bottom layer [box (3)]. In the bottom layer, after the cap is adjusted, MPEC/EPEC are solved to obtain equilibrium bidding strategies of the generators for each year of the planning horizon T
[boxes (6) and (7)]. The cap is considered to remain constant for
the first d years [box (4)] and decreasing thereafter [box (5)]. For
a given factor level combination, once the equilibrium strategies
are obtained for each year of the planning horizon T, the performance measures are updated [box (8)]. Once all years of the planning horizon are considered [box (9)], performance measures for
the complete horizon for the current factor level combination are
sent to ANOVA [box (10)] and the next combination is drawn for
evaluation. When all factor combinations are exhausted [box (2)],
ANOVA is performed [box (10)]. Regression equations (one for each
measure of performance) are developed using ANOVA results and
are sent as input for the multi-objective optimization [box (11)],
which yields the Pareto optimal C&T policies. In what follows, we
present the bi-level optimization model and discuss its solution
using the MPEC approach. Thereafter, we explain how EPEC approach is used in obtaining the equilibrium bidding strategies for
all generators in the electricity and allowance markets.
3.2. A bi-level model for joint allowance and electricity settlement
We adopt the bi-level framework that is commonly used in
modeling generator bidding behavior in deregulated electricity
markets [35–37]. We extend the objective functions of both levels
by incorporating emissions penalty cost in the upper level (see (2))
and both allowances revenue and social cost of carbon in the lower
level (see (6)). We also expand the constraints set in the lower level
(DC-OPF) to accommodate a C&T policy, which are explained later.
The solution of the modified DC-OPF yields both the electricity dispatch and the allowances allocation.
We consider that the generators and the consumers bid with
their linear supply and demand functions, respectively. ISO determines energy dispatch and allowances allocation by maximizing
social welfare while satisfying network and C&T constraints. The
flow in the network is controlled by the Kirchhoff’s law represented by the power transfer distribution factors (PTDFs). Capacity
limit constraints on the lines are also considered. The supply cost
of the generators, indexed by i, are assumed to be quadratic convex
functions given by C i ðqi Þ ¼ ai qi þ bi q2i , where ai and bi represent
the intercept and the slope of the supply function, respectively.
Consumers, indexed by j, are considered to have negative benefit
functions given by Dj ðqj Þ ¼ dj qj bj q2j ; qj 0 [38]. Generators
are paid at their marginal cost,
dC i ðqi Þ
@qi
¼ ai þ 2bi qi . Therefore, the
profit of a generator i is given as ðai þ 2bi qi Þqi ðAi qi þ Bi q2i Þ, where
Ai and Bi are the true cost parameters of generator i estimated
through a standard Brownian motion model as follows. For any
year t; Ati is obtained as Ati ¼ sAt1
þ rAt1
h, where s represent a
i
i
trend parameter, r represent the standard deviation of the process,
and h is a standard normal random variable. Our DC-OPF model is
similar to that presented by Hu and Ralph [38], which showed that
the use of the above functions ensure a unique solution. For some
other variants of the OPF model, readers are referred to Berry et al.
[39], Borenstein et al. [40] and Limpaitoon et al. [2].
Without loss of generality, we assume that the generators pass
on the cost of allowance to the consumers by modifying (shifting
up) the intercept (ai ) of their supply functions as a^i ¼ ai þ ci xi ,
where ci is the CO2 emissions factor and xi is the allowance bid
(cost) of generator i. Hence, the cost of allowance (xi ) does not
Fig. 1. A schematic of the solution methodology for the two-layer model.
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
explicitly appear in the generator’s (upper level) objective function.
Note that, shifting of the supply function results in a new equilibrium point, where consumption (dispatch) is reduced and electricity price is increased (see Fig. 2). Hence, xi implicitly impacts the
generator’s profit through quantity dispatch qi and excess emissions penalty paid by the generators.
Social welfare is commonly considered in the literature to be total benefit to the consumers Dj ðÞ minus total cost to the generators
C i ðÞ. Since allowances cost is passed onto the consumers, we include the cost of allowance as part of generators cost (i.e., use a^i
as the intercept of the supply cost function). We also add the
allowance revenue to social welfare based on the assumption that
it is recycled back as credits to consumers (to mitigate economic
impacts of increased electricity prices) and subsidies to green generators. Revenue recycling has been widely discussed and recommended by many economists [e.g., 41–43]. We also subtract from
the welfare function the social cost of the emissions, which is obtained by adding for all generators the product of the emissions
quantities and the social cost of carbon.
Let I ¼ f1; . . . ; ng be the set of generator, and J ¼ f1; . . . ; mg be
the set of consumers on the network. Let U ¼ ð/1 ; . . . ; /n Þ be the
vector of generators’ electricity bids, where /i ¼ ðai ; bi Þ represents
the bid of generator i, and X ¼ ðx1 ; . . . ; xn Þ be the vector of allowance bids. Let Q ¼ ðq1 ; . . . ; qN Þ, where N ¼ jðI [ JÞj, be the electricity
dispatch response vector obtained from the DC-OPF. Let
C ¼ ðc1 ; . . . ; cn Þ be the vector of CO2 emissions factor
ðtCO2 =MW hÞ of the generators. Then the yearly emissions of generator i is given by ci qi . Let H ¼ ðh1 ; . . . ; hn Þ be the vector of yearly
allowance allocation to the generators. Therefore, given the supply
function bids and the allowance bids of all other generators, Ui
and Xi , respectively, the bi-level optimization model for generator
i is given as follows.
Max ðai þ 2bi qi Þqi ðAi qi þ Bi q2i Þ pðci qi hi Þ
ð2Þ
ai ;xi
s:t: ai 2 ½A; A;
ð3Þ
xi 2 ½W; W;
ai þ ci xi 6 dj ; 8j; j ¼ 1; . . . ; J;
"
Q ; H ¼ Max
q;h
ð4Þ
X
X
X
X
Dj ðqÞ C i ðqÞ þ
xi h i q i c i p
j
s:t C l 6
X
i
i
qi ukl 6 C l ;
#
ð5Þ
ð6Þ
i
l ¼ 1; . . . ; L; ðl ; þl Þ
ð7Þ
k2ði;jÞ
X
X
qi þ
qj ¼ 0;
i
c
ðlÞ
ð8Þ
j
X
hi P 0;
ðkÞ
ð9Þ
i
ci qi hi P 0; 8i 2 I; ðqi Þ
qi Rlo P 0; 8i 2 I; ðsi Þ
Rup qi P 0; 8i 2 I; ðti Þ
qi P 0; 8i 2 I; ðpi Þ
qj P 0; 8j 2 J; ðji Þ
hi P 0;
8i 2 I: ðfi Þ
ð10Þ
ð11Þ
ð12Þ
ð13Þ
ð14Þ
ð15Þ
In the formulation, the elements within parentheses in constraints (7)–(15) represent the corresponding dual variables or
shadow prices. Our attention is focused primarily on l, which represents the system marginal energy price (cost) of the network [2].
Constraints (3) and (4) incorporate the bounds for electricity and
allowance bids, respectively. Constraint (5) ensures that the supply
function intersects with the demand curve. Objective function (6)
represent the social welfare where C i ðqÞ is the cost to the generators obtained using the shifted supply function (a^i ; bi ), Dj ðqÞ is
the benefit to the consumers, and p denotes to the SCC. Flow con-
P(q)
Shifted supply function
New equilibrium
Original supply function
(
+
)
Original equilibrium
Demand function
(
b)
q
Fig. 2. Electricity market equilibrium with supply function shifted by allowance
cost.
straints are given in (7), where C l is the line capacity and ukl is the
PTDF for node k and line l. Energy balance is maintained by constraint (8). Constraint (9) ensures that the allocation of allowances
does not violate the cap. Since we do not consider banking of
allowances, constraint Eq. (10) ensures that a generator is not allocated with more allowances than the emissions emitted. Maximum and minimum production level for each generator are
controlled by constraints (11) and (12) respectively. Finally, (13)–
(15) are non-negativity constraints for electricity dispatch and
allowance allocation.
3.3. Model solution for equilibrium bidding strategies: A MPEC/EPEC
approach
Bi-level optimization models have been widely studied in the
literature [44,45]. Bi-level models include two mathematical programs, where one serves as a constraint for the other. For the lower
level problem, with a convex objective function and non-empty
feasible set, the first order necessary conditions for a solution to
be optimal are given (under some regularity conditions) by the
Karush Kuhn Tucker (KKT) equations. Hence, replacing the lower level problem in Section 3.2 by the set of optimality conditions,
yields what is known as a mathematical program with equilibrium
constraints (MPEC). Further details on the MPEC models can be
found in [46–48].
In the other hand, an equilibrium program with equilibrium
constraint (EPEC) is defined as a game, EPEC ¼ fðMPECÞgn1 , among
competing generators. Typically, MPECs have non-convex feasible
sets (due to the complementarity constraints), therefore the resulting games are likely to have non-convex feasible strategy sets.
Since a global equilibrium for this type of games is difficult to identify, a local Nash equilibrium concept is presented in [38]. The local
Nash equilibrium is comprised of stationary points of the MPEC
problem for each player i in the game EPEC ¼ fðMPECÞgn1 . Lack of
knowledge of other players’ exact strategies, which might also be
changing with time, along with other uncertainties may minimize
the value of the effort required to seek global optimal strategies.
Literature presents different strategies to solve EPECs, of which
linear and non-linear complementarity (LCP/NCP) formulation [49]
and diagonalization methods [50,51] are the most discussed. In
this paper we consider a diagonalization method algorithm as presented in [52]. In the diagonalization method, MPECs for all generators are solved, for which we use a regularization method in
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
conjunction with NLP solvers [53,54]. For other techniques to solve
MPECs, readers are refers to [55,47].
3.4. Multi-objective optimization for Pareto optimal designs
In a complex system with multiple performance measures and
the set of significant design factors that affect them, the nature
of the relationship could vary widely among the performance measures. That is, a set of factor values (levels) that are optimal for a
particular performance measure may not be optimal for other
measures. For example, the C&T and network parameters that minimizes CO2 emissions may not yield lowest electricity prices, when
both reduced emissions and lower prices may be among the priorities. Therefore, a design approach that can balances among multiple priorities must be considered. Multi-objective optimization
models, that yield Pareto fronts, provide such an approach.
A Pareto front is a set of points representing factor level combinations where all points are Pareto efficient. A Pareto efficient
point indicates that no measure of performance can be further improved without worsening one or more of the other performance
measures. Similarly, for any point outside of the Pareto front, by
moving the point onto the front, one or more measures can be improved without worsening the others. In order to obtain these
points, we formulate the multi-objective optimization problem
(16). We used the NSGA-II genetic algorithm [57] to solve the model and obtain the Pareto envelope. Equations CL (consumption level), TE (total emissions), and AMP (average marginal price) are
explained in detail in Section 4.3.1.
Min TEðc; r; p; b; p; lÞ;
Max CLðc; r; p; b; p; lÞ;
ð16Þ
Min AMPðc; r; p; b; p; lÞ;
s:t: bounds on c; r; p; b; p; l:
4. C&T policy development for a sample network
In this section, we demonstrate, using a sample electricity network, how our model can be used to develop Pareto optimal C&T
policies. We conduct the numerical study in two phases. We first
implement the bottom layer of the model for evaluating a number
of different ad hoc C&T policy scenarios, as shown in Table 1. In the
ad hoc policies, penalty and the cap vary within their corresponding ranges (per RGGI), and SCC varies according to the IWG recommendations (see Section 2). The above variations coupled with
different levels of demand-price sensitivity (slope of the demand
curve) are captured as the three main scenarios SN1, SN2, and
SN3 (see Table 1). Samples of results obtained by evaluating these
scenarios are presented to demonstrate the impact of both C&T
policy and network parameters. In phase 2, we implement the
complete two layer model. We first develop a factorial experiment
and conduct ANOVA. Response surfaces derived from ANOVA are
used to obtain Pareto optimal C&T policies via multi-objective optimization model.
4.1. Sample electricity network
We consider a sample network with 4-nodes and 5-lines (see
Fig. 3). Similar sample networks were considered in numerical
studies in [58–62]. The network operates under a CO2 C&T policy
with three generation nodes and one load node. The network has
two fossil fuel (coal) generators (GENCO1 and GENCO3) and one
green generator (GENCO2). The green generator does not participate in the allowance auction. Emissions factor for both coal generators is assume to be one (i.e., 1 ton of CO2 emission per MW h
of electricity production). Allowances are distributed among competing generators using a discriminatory (pay-as-bid) auction pricing strategy. The cap is considered to decrease at a yearly rate of
2.5% starting the sixth year of implementation (this is similar to
the policy implemented by the RGGI). Line capacities for C1 and
C4 are set to 80 MW h, while the rest of the lines have a capacity
of 120 MW h. The model is implemented for a thirty year planning
horizon. Demand is considered to increase at a yearly rate of 1.1%,
which is implemented in our model by raising the intercept of the
demand function.
For supply functions, we consider that the generators bid only
on the intercept parameter a. Intercept parameter Ai of the true
cost function of the generators is estimated using a Brownian motion model (BMM) as explained in Section 3.2. The trend parameter
(s) of the BMM model is set to 0.0059 for green generator and
0.0662 for coal generators. The standard deviation (r) of the
BMM model is set to 0.081 for green generator and 0.0714 for coal
generators. The above numerical values were obtained from the
Electric Power Annual Data 2009 ([63]). For year t ¼ 1; A1i is set
to 2011 cost given in the above data. For all generators, we set
the value of Bi to 0.05.
4.2. Analysis of ad hoc C&T policies
We implemented the policy evaluation part of our model in
GAMS using MPEC and CONOPT3 solvers. For each scenario in Table 1, we obtained the equilibrium bidding strategies for the complete planning horizon. For each year of the planning horizon, we
recorded performance measures such as production levels of each
generator (sum of which is the consumption level), marginal electricity price, generator profits, emission levels, and market share of
the green generator. These yearly measures are used in calculating
the network performance measures for the complete horizon.
4.2.1. Impact of C&T and network parameters on performance
measures
In this section we examine the effect (sensitivity) of some of the
policy and network parameters on the performance measures. A
more comprehensive analysis of the impact of the C&T and network parameters on the network performance is conducted in Section 4.3.1 using the analysis of variance (ANOVA) technique.
Tables 2–4 summarize the performance measures for various ad
hoc C&T policies belonging to SN1, SN2, and SN3 with a cap of 80
tCO2 . It can be observed that a higher sensitivity in the price reduces production (or consumption) and emission levels. Social cost
of carbon further contributes to this reduction. Along with the
reduction of emissions, the percentage of market share of green
Table 1
Ad hoc C&T policies.
Scenario designation
Demand (b) (slope)
Penalty (p) ð$=tCO2 Þ
SCC (p) ð$=tCO2 Þ
Cap (c) (tCO2 )
SN1
SN2
SN3
0.025
0.05
0.075
10–50
10–50
10–50
0–21
0–21
0–21
80–120
80–120
80–120
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
4.2.1.1. Generators behavior. Fig. 4 shows all three generators’
yearly production levels for the scenario SN1 under two different
penalty levels ð$30=tCO2 and $50=tCO2 ) and zero SCC. Recall that
we considered the emissions factor c ¼ 1, which means each
MW h of coal based electricity production results in one unit
(ton) of CO2 emission. Hence, sum of the production levels of
GENCO1 and GENCO3 represents the total emission in the network.
GENCO2 is the green generator with zero emissions. The thicker
solid lines in the figures represent the total emissions cap for the
network.
In the low penalty cost scenario (p ¼ $30=tCO2 ), as the demand
grows over the years and the cap reduces, the coal generators still
find it profitable to increase their share of production (above green
generation) while paying more in penalties. Whereas, in the high
penalty cost scenario (p ¼ $50=tCO2 ), the coal generators control
their bids to significantly lower their production. In fact, till year
22, the combined emissions from GENCO 1 and 3 remain below
the cap. The green generator maintains a high level of production.
Fig. 3. A 4-node sample network for numerical study.
sources is in general higher when consumers are more sensitive,
and their profits are slightly reduced. Electricity prices suffer a significant reduction of almost $10=MW h on average.
Table 2
Performance of ad hoc C&T policies belonging to SN1 (demand slope 0.025).
SCC-0
Production (MW h)
Emissions (tCO2 )
Market share (%)
AMP ($/MW h)
Profit ($)
SCC-21
10
30
50
10
30
50
8021
5633
32
66.7
129,436
8023
5641
36
67.9
128,290
7483
4038
45
67.4
165,259
8026
5636
38
67.8
128,470
7397
4018
55
68.2
167,445
5893
2057
57
69.8
203,593
Table 3
Performance of ad hoc C&T policies belonging to SN2 (demand slope 0.05).
SCC-0
Production (MW h)
Emissions (tCO2 )
Market share (%)
AMP ($/MW h)
Profit ($)
SCC- 21
10
30
50
10
30
50
7742
5094
34
55.6
103,740
7617
5060
34
57.01
97,776
6155
2580
58
60.98
155,773
7738
4917
36
56.39
1,067,733
6305
2415
62
60.04
161,307
5942
2016
66
62.15
176,520
Table 4
Performance of ad hoc C&T policies belonging to SN3 (demand slope 0.075).
SCC-0
Production (MW h)
Emissions (tCO2 )
Market share (%)
AMP ($/MW h)
Profit ($)
SCC- 21
10
30
50
10
30
50
7983
5772
28
42.37
64,193
6754
3472
49
48.19
103,425
5575
1905
66
54.47
142,079
6672
3362
50
48.65
107,056
5694
2052
64
53.82
139,902
5641
1982
65
54.12
141,322
Production level (p=$30/tCO2, SCC=0)
Production level (p=$50/tCO2, SCC=0)
160
120
MWh (tCO2)
MWh (tCO2)
160
80
40
0
1
4
7
10
13
16
19
22
25
28
120
80
40
0
1
4
7
10
13
Year
GENCO1
GENCO2
16
19
22
25
28
Year
GENCO3
cap
GENCO1
GENCO2
Fig. 4. Production level comparison for SN1 under different penalty levels.
GENCO3
cap
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A higher cost of penalty causes the total electricity consumption
over the horizon to fall from 8256 MW h to 6213 MW h (a 23%
reduction) and the total emissions for the horizon to reduce from
5635 tCO2 to 2795 tCO2 (a 50% reduction). The share of green generation in the two penalty levels are 32% and 49%, respectively. The
average marginal electricity price rises from $52:7= MW h to
$60:8= MW h (a 15.4% increase).
The penalty level choice varies for different markets. California’s
C&T set a penalty of 4 allowances per ton not covered by allowances, and the reserve allowance price was set to $10.71 in 2013,
which is set to increase to $11.34 for 2014 [64]. In the case of EU
ETS, the penalty was set to 40 euros in the phase I, and increased
to 100 euros during phase II [25]. The effectiveness of penalty is affected by allowance prices and quantity (cap), among others. For
example, if allowance prices drop, as was the case in phase I in
the EU ETS, a low level of penalty will unlikely generate emissions
reduction since generators can either afford to buy allowances,
and/or pay a penalty for non-compliance.
4.2.1.2. Generation and emission levels. Fig. 5 shows the effect of SCC
on total coal-based generation (or, equivalently, total emissions) as
well as total green generation over the horizon for increasing values of penalty cost and demand-price sensitivity. As expected, at
higher values of penalty and demand-price sensitivity, the coalbased generation decreases. At higher SCC, the reduction in coalbased generation is sharper (plot b). Also at higher SCC and penalty
costs, the effect of increased demand-price sensitivity is reduced
(plot b). As far as green generation is concerned, it can be observed
(plot a) that higher demand-price sensitivity compounds the effect
of penalty in increasing generation. This effect is further pronounced when SCC is higher (plot b).
4.2.1.3. Electricity prices. Fig. 6 depicts the average marginal prices
under the same parameter combinations as in Fig. 5. It can be seen
that higher penalty produces higher price, which increases further
with increased SCC, as expected. We also note that, higher demand-price sensitivity significantly lowers the price at all levels
of penalty and SCC.
4.2.1.4. Generators market share. Fig. 7 shows the market share of
green and coal based energy, aggregated over the complete horizon. As expected, an increase in penalty (from $10=CO2 to
$50=tCO2 ) raises the share of green energy from 32% to 45%. The increase in green energy is further noticeable when the SCC is included. By considering SCC (Plot (b)) the share of green based
energy achieves a 57% versus a 43% of coal share for a penalty of
$50=tCO2 . Note that adding a SCC produces the share of green
based energy to be higher than coal based energy share (which is
not the case when SCC = 0). As mentioned in Section 2, SCC could
accelerate the transition to green technologies. Results presented
in this section helps to quantify the assertion.
(b) Total production/emissions SCC=$21/tCO2
Coal-0.025
Coal-0.05
Coal-0.075
Green-0.025
Penalty ($/tCO2)
MWh (tCO2)
MWh (tCO2)
(a) Total production/emissions SCC=$0/tCO2
Coal-0.025
Coal-0.05
Coal-0.075
Green-0.025
Green-0.05
Green-0.05
Green-0.075
Green-0.075
Penalty ($/tCO2)
Fig. 5. Impact of penalty and demand-price sensitivity on CO2 emissions (for different values of SCC).
Fig. 6. Impact of penalty and demand-price sensitivity on average marginal price of electricity (for different values of SCC).
(a) Market share (SCC=$0/tCO2)
Penalty ($/tCO2)
(b) Market share (SCC=$21/tCO2)
Penalty ($/tCO2)
Fig. 7. Impact of penalty and SCC on market share of green generation.
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(a) Generators profit (SCC= $0/tCO2 )
$
$
(b) Green generators profit
Penalty ($/tCO2)
Penalty ($/tCO2)
Fig. 8. Generator profits for different penalties and SCC.
4.2.1.5. Generators profit. Fig. 8(a) shows the cumulative generator
profits (coal generators are combined as one) for different penalty
levels and zero SCC. When higher penalty for non-compliance is
considered, coal generators see their profit to be reduced (approximately 43%). Coal based generators reduce their electricity production in order to maintain emissions according with the
allowances allocated. It was also observed in Figs. 5 and 7 that
the increase of penalty reduces the emissions (and hence production) and coal market share, resulting in lower profits for coal
based generators. Green generators observe an increase in profit
as we consider higher penalties (approximately 21%). It can be observed that including SCC (Plot (b)) further increases the cumulative profit of the green generator. These results show that C&T
policies will likely have conflicting implications. On one hand,
the environment and society may benefit from a reduction in emissions and increase in green power. On the other hand, coal-based
generators do not have incentives to participate in the market because of profit loses. This is one of the major concerns, for example
in the U.S., where fossil fuel energy sources are still a major portion
of the energy portfolio.
4.3. Development of Pareto optimal C&T policies
In this section, we first develop response surface equations for
each performance measure using a factorial design and ANOVA.
We then demonstrate how those equations can be used to obtain
Pareto optimal C&T designs.
4.3.1. Performance response surface generation
In order to obtain the response surfaces for the network performance measures, we adopt a designed factorial experiment comprising six factors: penalty, cap size, cap reduction rate, SCC,
demand-price sensitivity, and line capacities (congestion). We consider each factor at three levels resulting in a 36 factorial experiment. The numerical levels of the factors are given in Table 5. All
729 factor combinations were evaluated using the bottom layer
model, and the resulting performance measures were used to conduct ANOVA (with type I error a ¼ 0:05) utilizing the R software
(version 2.15.1). Results from ANOVA conducted separately for
each of the three performance measures (total CO2 emissions
(TE), total electricity consumption (CL), and average marginal price
Table 5
Factor levels for 36 experiment.
Factors
Cap reduction (r) (%)
Initial cap size (c)
Penalty (p)
Demand-price sensitivity (b)
Congestion (l)
SCC ðpÞ
(AMP)) are presented below. For more detailed information about
factorial design of experiments and ANOVA, we refer the readers
to [56].
4.3.1.1. Total CO2 Emissions Level: TE. When TE was used as the response variable, all main factors and some two level interactions
were found to be statistically significant. Using the significant factors and interactions, a second order regression model was developed. The model has a multiple R-squared value of 0.839 and an
adjusted R-squared value of 0.833.
TE ¼ 3696:63 1186:03p 989:66p þ 618:84b þ 683:56c
322:92l 316:37r þ 270:66p2 þ 353:8p2 253:34b
2
þ 648:22pp þ 340:81cp þ 256:36pc 219:83bl 217:25pb
þ 194:49pl þ 184:02lp 142:89rp 141:85pr 128:33bc
124:11bp þ 259:39p2 p 312:67p2 p2 þ 161:21pp2
2
150:1cp2 þ 234:56p2 b þ 61:19cr:
Factor effect plots for the six significant main factors are presented in Fig. 9. All the factors exhibit approximately linear behavior over the three levels, indicating that their impact on TE are
either higher the better or lower the better.
4.3.1.2. Total Consumption Level: CL. As in the case for TE, all six
main factors and some two level interactions were found to be statistically significant. A second order regression model incorporating the significant factors and interactions was developed. The
model has a multiple R-squared value of 0.842 and an adjusted
R-squared value of 0.835. Plots of the factor effects at various levels
are presented in Fig. 10. Non-linearity in the factor effects can be
observed to be a bit higher than in the case of TE, which can also
be seen in the regression equation that has more non-linear terms.
CL ¼ 7315:88 1237:44p þ 689:52b 758:85p þ 425:21c
2
263:04l 521:34p2 344:16b 232:83r þ 86:63l
2
þ 84:57c2 þ 67:39p2 þ 304:15pp 216:74bl þ 167:3pl
166:88bp þ 157:77pc þ 156:99bp þ 146:67lp þ 128:88cp
2
þ 128:26cr þ 292:63p2 b 91:97pr þ 112:51pp2
2
111:2cp2 60:14r p þ 144:98rc2 þ 55:24cl þ 95:2bl
2
2
2
þ 92:89cl 52:3bc þ 90:43pl þ 84:05rl þ 244:67p2 p
89:2cp2 :
Coded values
1
0
1
2.5
80
10
0.075
120
0
5
120
30
0.05
100
21
7.5
160
50
0.025
80
42
4.3.1.3. Average Marginal System Price: AMP. For AMP as the
response variable, the only main factor that was not statistically
significant was the cap reduction rate. Some of the two level interactions were significant. The regression model has a multiple
R-squared value of 0.637 and an adjusted R-squared value of
0.621. Lower R-squared value is, perhaps, indicative of the fact that
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F. Feijoo, T.K. Das / Applied Energy 119 (2014) 371–383
Fig. 9. Factor effect plots for total emissions.
Factor effects plot for consumption (b)
Penalty
MWh
MWh
Factor effects plot for consumption (a)
Cap reduction
Demand Slope
Cap size
SCC
Line capacity
Factor levels
Factor levels
Fig. 10. Factor effect plots for consumption level.
Fig. 11. Factor effect plots for average electricity marginal price.
the factor effects have a higher order (>2) of non-linearity (see
Fig. 11), which could not be captured from a 3-level factorial
experiment.
AMP ¼ 55:1717 þ 6:809p þ 2:9673p2 þ 1:6172p þ 0:9829p2
2
2
þ 5:2204b þ 3:6239b 1:1398c 0:3786l 3:2884l
0:2442r 0:6737r 2 0:3548c2 0:8224pp 1:0853bp
2
þ 0:8359r p þ 0:7443lp 3:5386p2 b þ 2:4959p2 l
2
2:0189p2 p þ 2:3979p2 b þ 1:2676p2 r 0:7284pc
0:7782pl 0:8208br 1:1743cr 0:7409lr 1:3341cr 2
1:2651rc2 1:3376cl:
4.4. Design of Pareto optimal C&T policies
We first demonstrate the need for Pareto optimal designs to
simultaneously accommodate multiple performance measures.
Table 6
Performance measures of C&T policies optimized for individual measures.
Optimized
measure
Consumption
(MW h)
Emissions
ðtCO2 Þ
AMP
($/MW h)
CL
TE
AMP
8950
5606
7747
7272
765
4274
35.8
54.7
21.9
Thereafter, we discuss how to obtain the Pareto optimal C&T policy
designs.
The need for Pareto designs is motivated by the results presented in Table 6, which shows that a C&T policy optimized for a
particular performance measure tends to produce poor outcomes
for the other measures. Notice from the table that, a design optimized for TE yields a consumption level of 5606 MW h whereas
the design optimized for CL has a consumption level of
8950 MW h. Similarly, a design optimized for CL yields an AMP
of $35.8/MW h, which is much higher than $21.9/MW h which is
attained by a design optimizing AMP.
We formulated a multi-objective optimization model, as presented earlier in (16), using the equations developed above for
TE, CL, and AMP. Of the six variables in these equations, SCC (p)
and demand-price sensitivity (b) are not controlled by a C&T policy
maker, and hence each of those variables were considered at three
different fixed values, resulting in nine possible combinations. For
each of these nine combinations, the multi-objective model was
solved and the optimal settings for the other four variables (cap
reduction rate, cap size, penalty, and congestion level) were
obtained.
Before presenting the complete Pareto envelopes for all three
measures, parts (a), (b), and (c) of Fig. 12 show the Pareto fronts
for each pair of performance measures. Each front is plotted for
three combinations of coded values of ðp; bÞ (0, 1), (0, 0), and
(0, 1). It may be noted that, for a fixed SCC value of 21 (coded value
of 0), lower values of demand-price sensitivity result in higher Par-
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Consider the scenario with p ¼ 0 and b = 1. Say that it is desired
to have an aggregated consumption level above 9000 MW h and
the total emissions below 6000 tCO2 over the complete planning
horizon. It can be seen from the Table 8 that the Pareto efficient
designs 85, 86, 87, 90, and 91 are those that meet both the consumption and emissions conditions. Since AMP is lower the better,
the Pareto efficient design with the lowest AMP should be chosen.
The design 86 happens to be the one with lowest AMP of $56.17/
MW h, for which the actual consumption level is 9017.7 MW h
and the emissions is 5211.57 tCO2 . The values of the C&T
design parameters for the Pareto efficient point 86 are
p ¼ $28:2=tCO2 ; r ¼ 0:075, c ¼ 159:05 tCO2 , and l = 120 MW h.
Table 7
Average values of performance measures from Pareto envelopes.
p
b
CL
TE
AMP
1
1
1
0
0
0
0
0
0
1
0
1
1
0
1
1
0
1
7125.03
7850.56
8505.91
6314.70
7425.11
7774.23
4942.74
5888.57
6699.43
3921.19
4788.07
5689.47
2907.43
3760.65
4007.63
2620.05
2883.47
3232.42
43.79
50.18
57.51
50.17
52.41
61.64
57.53
63.02
67.95
eto fronts and therefore increases the levels of consumption, emissions, and AMP.
We obtained the Pareto envelopes (considering all three measures) for all nine combinations of p and b. For each envelope,
we computed the average values of the measures over all the Pareto efficient points. In the solution of the multi-objective model,
we chose to obtain 100 Pareto efficient points. The results are presented in Table 7. As observed earlier in Fig. 12, for fixed value of p,
decrease in demand-price sensitivity (from 1 to 1) results in
increase in CL, TE, and AMP. For a fixed value of b, increase in
SCC results in decreases in CL and TE and an increase in AMP.
To illustrate further, we present in Fig. 13 two complete Pareto
envelopes for two arbitrarily chosen combination of ðp; bÞ of (0, 0)
and (0, 1). It can be observed that Pareto efficient designs that minimize emissions yield lower consumption levels and higher average marginal prices. Allowing higher emissions results in
increase in consumption and decrease in price. This shows that
stricter emissions policies will increase electricity prices. For sixty
of the one hundred efficient points on the Pareto envelope (0, 1),
the corresponding values of the four C&T design parameters are
shown in Table 8. We picked a subset of the designs (instead of
all 100) for reasons of space. In what follows, we describe how
an envelope can be used to select an appropriate design for a
C&T policy.
Pareto front for production and emissions
5. Concluding remarks
In this paper, we have developed a mathematical–statistical
model that allows us to obtain Pareto optimal C&T policies.
The model has two broad layers. The bottom (policy evaluation)
layer evaluates the impact of C&T and network parameters on
the performance measures of an electricity network. The top
(policy optimization) layer obtains the Pareto optimal designs
using the results of the bottom layer. The existing literature,
containing both empirical studies [8] and mathematical models
[2,12], helps to effectively evaluate the impact of given C&T policies. Our research extends the literature from evaluation of C&T
policies to design of Pareto optimal policies that accommodate
different interests of the network constituents (e.g., higher consumption, lower emissions, lower electricity prices). We have
demonstrated that a Pareto envelope generated by our model
can serve as an useful tool for policy makers to select alternative
C&T policies satisfying various interests of the electricity network constituents. Results presented in the paper also examine
the sensitivity of important factors affecting electricity markets
such as social cost of carbon and demand-price sensitivity. Electricity generators can benefit from this model by using the bottom layer to assess impact of given C&T policies on their bidding
strategies and capacity expansion planning.
AMP ($/MWh)
Pareto front for emissions and AMP
AMP ($/MWh)
Emissions (tCO2)
Pareto front for production and AMP
Production (MWh)
Production (MWh)
Pareto Front (π=0,b=-1)
Pareto Front (π=0,b=1)
Pareto Front (π=0,b=0)
Pareto Front (π=0,b=-1)
Pareto Front (π=0,b=1)
Emissions (tCO2)
Pareto Front (π=0,b=0)
Pareto Front (π=0,b=-1)
Pareto Front (π=0,b=0)
Pareto Front (π=0,b=1)
Fig. 12. Pareto fronts for each possible performance pair.
Pareto front ( π=0,b=1) (b)
Pareto efficient designs
Pareto efficient designs
Consumption
Emissions
$/MWh
$/MWh
MWh (tCO2)
MWh (tCO2)
Pareto front ( π=0,b=0) (a)
AMP
Fig. 13. Pareto envelopes.
Consumption
Emissions
AMP
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Table 8
Pareto C&T designs for
Design
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
p=0 and b = 1.
Coded values
Real values
Performance measures
p
r
c
1
Penalty
Cap red.
Cap
Lines cap
Consumption
Emissions
AMP
l.00
l.00
l.00
l.00
l.00
l.00
0.91
l.00
0.97
l.00
l.00
l.00
l.00
l.00
l.00
l.00
l.00
0.91
l.00
l.00
l.00
0.98
l.00
l.00
l.00
l.00
l.00
l.00
0.00
l.00
0.07
0.79
0.25
0.76
0.76
0.35
0.65
0.79
0.76
0.69
0.76
0.79
0.76
0.76
0.24
0.24
0.99
0.99
0.99
0.76
0.99
0.96
0.69
0.93
0.98
0.93
0.98
0.99
0.99
0.99
l.00
l.00
l.00
l.00
0.75
l.00
0.96
l.00
l.00
l.00
0.91
0.97
0.94
0.40
0.91
0.00
l.00
0.49
0.91
0.97
0.50
0.91
0.91
0.97
0.91
0.91
l.00
0.78
0.97
0.50
l.00
0.79
l.00
0.95
0.93
l.00
0.94
0.79
0.58
0.96
0.93
0.94
0.43
0.43
0.75
l.00
0.99
0.99
0.84
l.00
0.93
0.13
0.96
0.93
0.87
0.93
0.87
0.01
0.01
0.99
l.00
0.91
l.00
l.00
l.00
l.00
l.00
0.51
0.51
l.00
0.92
0.72
0.31
l.00
0.25
0.96
0.06
0.28
0.45
0.72
0.03
0.45
0.28
0.29
0.01
0.03
0.51
0.63
0.75
0.65
l.00
0.64
0.62
l.00
0.72
0.56
0.49
0.99
0.15
0.56
0.75
0.71
0.76
0.76
0.98
0.98
0.33
0.27
0.27
0.74
0.88
0.76
0.88
0.77
0.78
0.76
0.98
0.99
0.96
0.99
0.71
0.65
0.90
0.99
0.96
0.21
0.90
0.71
0.77
0.07
0.07
0.ll
0.96
0.07
0.07
0.07
0.ll
0.90
0.48
0.61
0.07
0.58
0.58
0.99
0.48
0.58
0.97
0.10
0.90
0.07
0.60
0.37
0.85
0.51
0.61
0.85
0.58
0.ll
0.51
0.92
0.51
0.57
0.60
0.61
0.99
l.00
0.75
0.75
0.75
0.85
0.61
0.73
l.00
0.95
0.98
l.00
0.98
0.94
l.00
l.00
50.00
50.00
50.00
50.00
50.00
50.00
48.12
50.00
49.37
50.00
50.00
50.00
50.00
50.00
50.00
50.00
50.00
48.12
50.00
50.00
50.00
49.69
50.00
50.00
50.00
50.00
50.00
50.00
29.92
50.00
28.67
14.24
24.90
14.86
14.86
23.02
17.06
14.24
14.86
16.12
14.86
14.24
14.86
14.86
25.22
25.22
10.16
10.16
10.16
14.86
10.16
10.78
16.12
11.41
10.47
11.41
10.47
10.16
10.16
10.16
0.075
0.075
0.075
0.075
0.069
0.075
0.074
0.075
0.075
0.075
0.073
0.074
0.073
0.060
0.073
0.050
0.075
0.062
0.073
0.074
0.062
0.073
0.073
0.074
0.073
0.073
0.075
0.070
0.074
0.062
0.075
0.070
0.075
0.074
0.073
0.075
0.073
0.070
0.036
0.074
0.073
0.073
0.061
0.061
0.069
0.075
0.075
0.075
0.071
0.075
0.073
0.053
0.074
0.073
0.028
0.073
0.028
0.050
0.050
0.075
80.00
83.76
80.00
80.00
80.00
80.00
80.00
99.45
99.45
80.00
83.14
91.29
107.61
80.00
110.12
81.57
117.65
108.86
101.96
91.29
118.90
101.96
108.86
131.45
119.53
118.90
140.24
145.25
90.04
145.88
160.00
94.43
144.94
80.00
91.29
142.43
139.61
159.69
114.20
97.57
149.96
148.39
150.27
150.27
159.06
159.06
106.98
130.82
130.82
149.65
155.29
150.27
155.29
150.90
151.22
150.27
159.06
159.69
158.43
159.69
85.80
87.06
82.04
80.16
80.78
95.84
82.04
85.80
84.55
101.49
101.49
102.12
80.78
101.49
101.49
101.49
102.12
82.04
109.65
112.16
101.49
111.53
111.53
80.16
109.65
111.53
80.63
101.96
82.04
101.33
112.00
107.45
117.02
110.27
112.16
117.02
111.53
102.12
110.27
118.43
110.27
111.37
112.00
112.16
119.84
120.00
114.98
114.98
114.98
117.02
112.16
114.51
120.00
119.06
119.69
120.00
119.53
118.75
120.00
120.00
6137.88
6145.92
6148.38
6160.64
6181.69
6201.12
6213.98
6225.41
6254.46
6295.03
6313.72
6327.84
6363.68
6401.94
6438.88
6485.80
6490.29
6528.29
6587.54
6607.42
6654.26
6664.47
6698.67
6713.10
6735.35
6792.65
6862.97
6938.0l
6974.37
7012.55
8476.50
8521.54
8618.33
8631.73
8655.31
8675.17
8699.10
8755.75
8794.46
8823.88
8882.61
8938.72
8985.68
8992.19
9006.27
9017.70
9066.58
9142.15
9155.30
9187.42
9254.73
9285.63
9371.63
9478.67
9513.49
9521.13
9525.84
9571.73
9620.80
9716.60
1780.92
1870.79
1715.37
1682.59
1823.88
1955.74
1779.46
2132.34
2125.61
2054.07
2159.18
2284.80
2222.25
2363.68
2642.75
2601.45
2745.16
2518.58
2638.59
2459.61
2987.76
2682.77
2795.07
2625.98
2953.48
2975.0l
2779.06
3333.37
2813.35
3451.77
4769.34
5134.39
4954.14
5000.01
5165.07
5064.49
5354.30
5451.45
5618.38
5281.57
5587.20
5662.54
5730.99
5736.41
5262.84
5211.57
5915.05
6042.52
6067.47
5805.73
6077.10
6189.26
5859.68
6196.06
6539.19
6226.28
6554.68
6450.68
6492.06
6381.46
69.25
69.36
68.14
67.51
67.59
71.04
67.14
67.85
67.14
71.32
71.11
70.88
65.33
70.67
69.64
70.45
68.95
65.14
69.98
70.16
69.12
69.59
69.40
61.75
68.80
68.69
60.34
66.18
61.37
66.88
57.23
64.49
58.49
63.85
63.57
58.84
61.09
60.71
63.46
61.67
60.74
60.94
62.43
62.40
57.73
56.17
64.11
63.04
63.28
59.44
61.47
63.63
58.04
60.20
61.98
60.02
62.46
62.98
62.63
59.18
Computational challenges associated with our model are limited to the bottom layer (in particular, solving the EPEC problem),
especially when a network has a large number of generators [38].
The number of time periods in the planning horizon also adds to
the computational burden as EPEC needs to be solved for each period. However, the computation time for the top layer model is
independent of the size of the network, number of generators,
and the time periods of the planning horizon.
Acknowledgements
Authors acknowledge the useful comments and suggestions
provided by the reviewers which have significantly improved the
quality of the manuscript.
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